Edit: in short, it can avoid starvation, but isn't fully reliable, thanks to the screwy job acquisition code and because only the owner will ever consider giving food. Putting the animal into a pasture is a much better choice.
A pet not assigned to a pasture will follow its owner around everywhere. In this mode, a food-dependent pet will not graze and become hungry, until it either starves to death or the owner decides to feed it. I haven't bothered too much with it, but dwarfs do use proper food (can't remember if only raw-edible plants or prepared food, as well) and will usually only consider feeding their pets when they're well into 'starving'. If the owner is extremely busy or on break, this can lead to a starved pet. Whenever i watched, the owner was the only dwarf who ever thought of feeding the pet.
Since such pets will just follow the dwarf around, contributing to fortress traffic and offering nothing but a slightly increased chance of 'comforted by a pet' happy thoughts, it's generally better to just stick adopted grazers into a pasture; this way, they won't starve and won't get into the way of traffic. And it's generally best not to allow grazers for adoption in the first place; you can't prevent them as immigrant pets, but there's no need having additional non-slaughterable bulls or like nonsense wasting pasture space.